Limit Theorem for the Riemann Zeta-Function in the Space of Continuous Functions

Abstract

Let C _∞ = C ∪ {∞} be the Riemann sphere and let d(s ₁, s ₂) be a metric on C _∞ given by the formulae

$$d({s_1},{s_2}) = \frac{{2\left| {{s_1} - {s_2}} \right|}}{{\sqrt {1 + {{\left| {{s_1}} \right|}^2}} \sqrt {1 + {{\left| {{s_2}} \right|}^2}} }},d(s,\infty ) = \frac{2}{{\sqrt {1 + {{\left| {{s_2}} \right|}^2}} }},d(\infty ,\infty ) = 0.$$